Some Questions

[Niwashi]

In terms of the amount of XP you get per item or quest, GC provisioning gives the most, then Ixal quests, then leves, then just regular grinding.
That's also the order of how limited they are, however, with the GC provisioning only once per class per day, Ixal 12 times (any class) per day, leves as many as you have allowances for, and grinding of course is unlimited.

GC Provisioning and Leves have the advantage that it's possible to just buy the items and turn them in. Sometimes that's too expensive to be a good idea, but other times you can find what you need cheap and get a quick effortless XP boost.

Ixal quests have the advantage that the only supplies you need to provide are the crafting shards you'll use (and most reward some shards in exchange, though they may or may not be the type you need). The other materials are collected by doing the quest itself.

Personally, I've found a combination of GC provisioning and Ixal quests the best approach for me, but many people prefer using lots of leves. If you're willing to spend lots of gil on it, leves can be the fastest. (The fact that I'm not willing to spend much gil on it may be why the leve approach doesn't work well for me. If you're gathering all the materials yourself to make all the items yourself to then turn in, leves become a bigger time sink than GC provisioning and Ixal quests.)

I am currently trying to level Miner (wish I could collect them with Botanist instead) as well so I can collect my own lightning shards/crystals.

You can collect any type of shard with either Botanist or Miner. (Though getting Miner to 20 to unlock the "Byregot's Ward" skill will help. Once unlocked, though, all six of those ward skills can be cross-classed, Miner to Botanist or Botanist to Miner, so they'll both be able to use all of them.)

[Sentaryuu]

You can collect any type of shard with either Botanist or Miner. (Though getting Miner to 20 to unlock the "Byregot's Ward" skill will help. Once unlocked, though, all six of those ward skills can be cross-classed, Miner to Botanist or Botanist to Miner, so they'll both be able to use all of them.)

Oh. I did not know this. Well now that just gives me the idea to just lv Miner to 20 and put that ability on Botanist since it's the one I'm wanting to lv more at the moment. Sister is also leveling Miner, which will help as well. Thank you!